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Kevin Ryde
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Math::NumSeq::LemoineCount -- number of representations as P+2*Q for primes P,Q


 use Math::NumSeq::LemoineCount;
 my $seq = Math::NumSeq::LemoineCount->new;
 my ($i, $value) = $seq->next;


This is a count of how many ways i can be represented as P+2*Q for primes P,Q, starting from i=1.

    0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 2, 1, 2, 0, 2, 1, 4, 0, ...
    starting i=1

For example i=6 can only be written 2+2*2 so just 1 way. But i=9 is 3+2*3=9 and 5+2*2=9 so 2 ways.

Odd Numbers

Option on_values => 'odd' gives the count on just the odd numbers, starting i=0 for number of ways "1" can be expressed (none),

    0, 0, 0, 1, 2, 2, 2, 2, 4, 2, 3, 3, 3, 4, 4, 2, 5, 3, 4, ...
    starting i=0

Lemoine conjectured circa 1894 that all odd i >= 7 can be represented as P+2*Q, which would be a count here always >=1.

Even Numbers

Even numbers i are not particularly interesting. An even number must have P even, ie. P=2, so i=2+2*Q for count

    count(even i) = 1 if i/2-1 is prime
                  = 0 if not


See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence classes.

$seq = Math::NumSeq::LemoineCount->new ()
$seq = Math::NumSeq::LemoineCount->new (on_values => 'odd')

Create and return a new sequence object.

Random Access

$value = $seq->ith($i)

Return the sequence value at $i, being the number of ways $i can be represented as P+2*Q for primes P,Q. or with the on_values=>'odd' option the number of ways for 2*$i+1.

This requires checking all primes up to $i or 2*$i+1 and the current code has a hard limit of 2**24 in the interests of not going into a near-infinite loop.


Math::NumSeq, Math::NumSeq::Primes, Math::NumSeq::GoldbachCount




Copyright 2012, 2013, 2014 Kevin Ryde

Math-NumSeq is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version.

Math-NumSeq is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with Math-NumSeq. If not, see <http://www.gnu.org/licenses/>.